So when I say that Brienne and Jaimes' story is an opera I mean in the sense of story-type, like Star Wars. Thus far we know how their stories can be parallel, or even mirrored.
Jaime is the handsome, firstborn son of one of the most powerful Lords in the realm. He grows up with an incredible interest and talent in fighting. He also admires great knights of esteem and honor, like Ser Arthur Dayne. But his great talent at such a young age made him arrogant. The Kingswood Brotherhood incident tests him, and he succeeds. Soon he becomes a knight of the Kingsguard, a position he quickly finds is not the majestic, mythical one he believed it to be. And then he kills Aerys, an event that placed him at odds with so many opposing vows, beliefs, and desires. A vow to protect the weak, protect his family and obey his father, a vow to protect and serve the King. He believes the people need protection, as Arthur Dayne protected and aided the people of the Kingswood. He believes his Father is a good man, despite his sometimes harsh justice. He desires to fulfill and live by his vows, all of them, but he cannot. And the reaction by everyone, excepting his father, is that Jaime is a terrible oath breaker who should have died protecting his King, despite the fact that it was true honor that drove Jaime to kill Aerys. He's gone all this time jaded more and more by this attitude against him, until he begins to live by his reputation of being an immoral, dishonorable knight. His captivity and crippling changes something in him, very small at first, but it begins to grow the more he's around Brienne, who struggles to live up to the same standard that once Jaime also strove to uphold. And so he rescues her from the bear pit, and sends her off to find and protect Sansa and Arya Stark in an effort to make good on his final oath to Catelyn Stark. From then on, he struggles to balance his vows to his family, and his vows to Stark. This balancing act culminates in his successful, bloodless retaking of Riverrun. And then we leave off with him setting out with Brienne of Tarth, once again, on a quest to save the Stark girls (or so he believes).
Brienne is the tall, muscular, masculine, "ugly" daughter of Tarth. She took to fighting, not courtly duties, and filled her head with the same myths and legends that romanticize Knighthood as Jaime believed so strongly as a child. Only she's a woman, and one who is terribly clumsy as a socialite. She failed miserably at trying to fill the role of a typical woman, and so her father relented and allowed her to train like a knight. After the death of King Robert the only man who was ever accepting of her, Lord Renly Baratheon, declares himself King and calls upon swords to join his union with Highgarden. Despite this being an act of treason, Brienne sets out to swear her sword to serve him. And when she shows skill in melee, he names her to his personal guard, an act of honor that deeply effects her. It is only so much more devastating to her, then, when she fails to protect him as he's murdered by a shadow, an event that leaves her feeling broken inside until Brienne finds herself in the service of Catelyn Stark, a woman of honor who revives in Brienne her sense of duty and honor. Brienne is then given a task, escort Ser Jaime Lannister back to King's Landing in return for the release of Lady Catelyn's young daughters. Yet again she fails an oath by allowing Jaime to be captured and mutilated. Suffering from fever during his time recovering from his wounded arm, he tells her of his own failures as a knight, tells her honestly why he killed Aerys and "soiled" his honor. This moment changes the way she sees Jaime, sees how nuanced he is, and how deep down he is not the dishonorable knight the realm has made him to be. This is reinforced when he risks his own life to save her from the bear pit, and Jaime begins to become an object of admiration for her much the same say Renly was. Despite all the japing and harsh comments, Jaime was honest and candid with her, and saved her life. Upon arriving at King's Landing, Brienne learns that she can never return the Stark girls to their lady mother. The Stark girls are not in King's Landing, and Lady Catelyn has been brutally murdered along with her son Robb. Yet Jaime intends to keep his oath to Stark, and sends her off with Oathkeeper to find Sansa and Arya. Her quest comes to an end when she is defeated and captured by the Brotherhood Without Banners, another oath failed. We leave off with her having been sent off to lure Jaime into a trap after her harrowing confrontation with the vengeful Lady Stoneheart.
That brings us to where we are with their stories.
The parallel in them is in how they each grew up with dreams of being great knights like that of Ser Arthur, and how those dreams have become tainted. Their arcs involving failed oaths and dishonor stand mirrored, and they share the same desire to find some way to fulfill their oaths. In Jaime's case, it is his oath to protect the innocent, the most honorable oath that he witnessed Arthur Dayne fulfilling in the Kingswood. This is reflected in his saving of Brienne, and his attempt to send her to find and protect Sansa and Arya Stark. In Brienne's case, it is her desire to Serve and Protect those honorable whom accept her service, even if it means her death. She would have done what Jaime could not do, had she been in his place with Aerys. And it's what she has failed to do all this time.
This sets up a play for tWoW involving their arcs, and what I suspect might be how their part of the story unfolds.
Brienne lures Jaime into the trap, as she was sent to do by Lady Stoneheart. He will be taken as their prisoner and brought before Lady Stoneheart. A trial is held, in which she accuses him of being an oathbreaker. Of this he admits some, but refutes others such as his success at taking Riverrun without taking up arms against the Tullys, and sending Brienne to find and protect the Stark girls. This also corroborates Brienne's insistence that she still holds to her oath to Lady Catelyn, and reminds Brienne of her honor. Lady Stoneheart, being who she is, denies that Jaime has honor and sentences him to death as an oathbreaker. Brienne then faces the crisis of conflicting oaths, just as Jaime had so long ago in Kings Landing. Either follow the command of the Lady Stoneheart, whom she had sworn service to, or protect Jaime with her life. In this moment, she chooses to protect Jaime, whom she sees as innocent of the crime for which Lady Stoneheart has judged him guilty. Swords are drawn, and Brienne fights with a fervor she has never fought with before. But the odds are against them, surrounded by enemies. She is able to kill Lem Lemoncloak and Lady Stoneheart, but not before being mortally wounded. She dies in Jaime's arms, having finally fulfilled her oath to die protecting the one she serves. The confrontation in the caves that results in Brienne's death, having saved Jaime, is the final event before Jaime makes the decision to take up his own destiny, and to live his life the way he chooses, to serve the cause he chooses. Down the line, I believe this would end with him returning to Kings Landing and eventually deciding to end Cersei's mad reign, particularly if she threatens the lives of the people if the city in a similar way Aerys did.
This possible scenario wraps up the story of Jaime and Brienne since it began in aCoK. That they have made an impact on each others' character is clear. I don't know how it will all unfold, but this is how I think it might go. I don't think Jaime's story will end until Daenerys Targaryen comes to take Kings Landing. He may even be the one who orders the city to open the Dany. This might bring Tyrion and Jaime face to face again.